Numerical solution of the fractional‐order logistic equation via the first‐kind Dickson polynomials and spectral tau method

Abstract

In this article, a numerical technique for solving numerically the fractional-order logistic equation (FLE) is presented. The fractional-order derivative of the studied problem is given through the Caputo operator of fractional derivatives. The first kind of Dickson polynomials is used as a basis for the desirable approximate solution. The presented technique is based on the operational matrices of these polynomials in both cases of integer and fractional-order derivatives. The handled problem will be transformed into a system of matrices that will be solved via an appropriate numerical technique. In detail, the convergence and error estimate of the suggested technique is discussed. Many numerical applications in several fields will be given to demonstrate the applicability, accuracy, and efficiency of the introduced technique. These numerical results will be compared to support our obtained theoretical results from one side and from another side to show the advantage of our method over some of the previous methods used in solving such applications.

Hybrid technique of conformal mapping and Chebyshev collocation method for solving time–space fractional order wave equation

“…However, the actual solution to many fractional‐order PDEs is more involved, and in some circumstances, it may be impossible to achieve. As a result, using approximate numerical solutions is incredibly important and vital 15–24 …”

Hybrid Fibonacci wavelet method to solve fractional‐order logistic growth model

“…The fractional logistic model is one of the mathematical models that has piqued the interest of several researchers [15]. The logistic (Verhulst) growth models have received extensive study and have been resolved using a variety of techniques, some of which are Legendre collocation methods [17], a collocation method for the numerical solution of fractional order [17], differential quadrature method [18], Dickson polynomial and spectral tau techniques [19], modified Eulerian numbers [20], Adomian decomposition method, and successive approximation method [7,8]. In general, it is quite difficult to solve fractional problems analytically, whether they are linear or nonlinear; therefore, it has been extremely helpful to generalize numerical approaches to estimate fractional derivatives and fractional differential equations.…”

“…The numerical outcomes demonstrated that the proposed technique is comparatively more efficient and convenient for handling different nonlinear problems. Our outcomes suggest that the novel approach is more effective and computationally convenient than traditional approaches [18, 20].…”

The Caputo–Fabrizio time-fractional Sharma–Tasso–Olver–Burgers equation and its valid approximations

“…Aydogan et al [7] explored the Rabies mathematical model in the CF sense and derived its approximate solutions through the Adomian decomposition method. For more studies, see [8][9][10][11][12][13][14][15][16][17][18][19][20].…”